Abstract

This paper extends Burckhardt’s solution for diffraction from a thick grating to include complex dielectric constants and nonsinusoidal stratifications. This allows any realistic periodic structure to be handled. Computed results are compared with coupled-wave theory, as described by Kogelnik, with emphasis on strongly absorbing gratings such as those made by photographing an interference pattern. Finally, some experimental holographic data are compared with computations that take into account the photographic nonlinearity between exposure and dielectric constant.

© 1973 Optical Society of America

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References

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  1. F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 58, 970 (1968).
    [CrossRef]
  2. F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).
    [CrossRef]
  3. K. Biedermann, Optik 28, 160 (1968).
  4. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [CrossRef]
  5. J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
    [CrossRef]
  6. R. L. Powell and Karl A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [CrossRef]
  7. J. S. Zelenka and J. R. Varner, Appl. Opt. 7, 2107 (1968).
    [CrossRef] [PubMed]
  8. C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).
    [CrossRef]
  9. H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
    [CrossRef]
  10. D. Kermisch, Ph.D. thesis, Polytechnic Institute of Brooklyn (1968).
  11. E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.
  12. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. P., Oxford, 1965), p. 24.
  13. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  14. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.
  15. R. L. Lamberts, J. Opt. Soc. Am. 60, 1389 (1970).
    [CrossRef]
  16. The relation between ∊i and exposure as expressed in Eq. (15) is assumed valid even when exposure is a function of position [recall that Eq. (15) was derived for a homogeneous medium] although, admittedly, transmittance as such is not well defined operationally for high-frequency patterns in thick emulsions. The problem arises from the difficulty of measuring a unique value of transmittance for high-frequency patterns in thick films. Different methods of illumination result in different measured values of T because of light scatter. At high spatial frequencies, Eq. (13) is no longer valid, but this does not invalidate the functional relation between ∊i; and exposure as derived from Eq. (15) and the D–log(exposure) curve. The quantity ∊i is uniquely defined; so long as its relation to exposure is independent of spatial frequency, then this relation as derived by use of Eq. (15) is correct.
  17. E. Leith, A. Kozma, J. Upatnieks, J. Marks, and N. Massey, Appl. Opt. 5, 1303 (1966).
    [CrossRef] [PubMed]
  18. H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972).
    [CrossRef]
  19. H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
    [CrossRef]
  20. I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

1972 (1)

1970 (1)

1969 (2)

1968 (5)

1967 (1)

1966 (3)

1965 (1)

Biedermann, K.

K. Biedermann, Optik 28, 160 (1968).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.

Burckhardt, C. B.

Edgett, C. D.

F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).
[CrossRef]

Goodman, J. W.

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

Kaspar, F. G.

F. G. Kaspar and R. L. Lamberts, J. Opt. Soc. Am. 58, 970 (1968).
[CrossRef]

F. G. Kaspar, R. L. Lamberts, and C. D. Edgett, J. Opt. Soc. Am. 53, 1289 (1968).
[CrossRef]

Kermisch, D.

D. Kermisch, Ph.D. thesis, Polytechnic Institute of Brooklyn (1968).

Knight, G. R.

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
[CrossRef]

H. Kogelnik, J. Opt. Soc. Am. 57, 431 (1967).
[CrossRef]

Kozma, A.

Lamberts, R. L.

Leith, E.

Marks, J.

Massey, N.

Powell, R. L.

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

Sanderson, R. L.

Smith, H. M.

Stetson, Karl A.

Streifer, W.

Upatnieks, J.

Varner, J. R.

Watson, G.

E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.

Whittaker, E.

E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. P., Oxford, 1965), p. 24.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.

Zelenka, J. S.

Appl. Opt. (3)

Bell System Tech. J. (1)

H. Kogelnik, Bell System Tech. J. 48, 2909 (1969).
[CrossRef]

J. Opt. Soc. Am. (9)

Optik (1)

K. Biedermann, Optik 28, 160 (1968).

Other (6)

D. Kermisch, Ph.D. thesis, Polytechnic Institute of Brooklyn (1968).

E. Whittaker and G. Watson, A Course in Modern Analysis, 4th ed. (Cambridge U. P., Cambridge, 1958), p. 413.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. P., Oxford, 1965), p. 24.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 613.

I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Integral No. 3.915-4, p. 482.

The relation between ∊i and exposure as expressed in Eq. (15) is assumed valid even when exposure is a function of position [recall that Eq. (15) was derived for a homogeneous medium] although, admittedly, transmittance as such is not well defined operationally for high-frequency patterns in thick emulsions. The problem arises from the difficulty of measuring a unique value of transmittance for high-frequency patterns in thick films. Different methods of illumination result in different measured values of T because of light scatter. At high spatial frequencies, Eq. (13) is no longer valid, but this does not invalidate the functional relation between ∊i; and exposure as derived from Eq. (15) and the D–log(exposure) curve. The quantity ∊i is uniquely defined; so long as its relation to exposure is independent of spatial frequency, then this relation as derived by use of Eq. (15) is correct.

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Figures (11)

Fig. 1
Fig. 1

Cross section of thick-grating model. Fundamental spatial frequency is b.

Fig. 2
Fig. 2

Diffraction efficiency (lower curve) and specular transmittance (upper curve) vs angle of incidence for pure absorption grating. Thickness equals 16 μm, modulation equals 50%, Bragg angle equals 30°.

Fig. 3
Fig. 3

Same as Fig. 2 except that model includes superimposed phase grating.

Fig. 4
Fig. 4

Same as Fig. 3 except for thickness equals 2μm.

Fig. 5
Fig. 5

Diffraction efficiency (lower two curves) and specular transmittance (upper curve) vs angle of incidence for combined phase–absorption grating. Thickness equals 8 μm, modulation equals 50%, Bragg angle equals 9°.

Fig. 6
Fig. 6

Same as Fig. 5 except that thickness equals 2 μm.

Fig. 7
Fig. 7

Comparison of thick-emulsion theory with data and thin-emulsion theory. Beam ratio is 90:1.

Fig. 8
Fig. 8

Same as Fig. 7 for beam ratio of 22:1.

Fig. 9
Fig. 9

Same as Fig. 7 for beam ratio of 7.6:1.

Fig. 10
Fig. 10

Amplitude transmittance T 1 2 vs relative exposure for Kodak spectroscopic plates, type 649F, developed five minutes in Kodak HRP developer at 68 °F.

Fig. 11
Fig. 11

Zero-order modified Bessel function, I0(x) vs x.

Equations (26)

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( 2 + k 2 ̂ ( x ) ) E ( x , z ) = 0 ,
̂ = n = 0 N ̂ n cos 2 π n b x .
E ( x , z ) = X ( x ) Z ( z ) ,
d 2 Z d z 2 = α 2 Z
d 2 X d x 2 + ( α 2 + k 2 n = 0 N ̂ n cos 2 π n b x ) X = 0 .
Z ( z ) = A e α z + A e α z .
X ( x ) = e 2 π i ξ x l = B l e 2 π i l b x .
ξ = sin θ λ .
( q 2 Λ ̂ 1 / 2 ̂ 2 / 2 ̂ 1 / 2 q 1 Λ ̂ 1 / 2 ̂ 2 / 2 ̂ 2 / 2 ̂ 1 / 2 q 0 Λ ̂ 1 / 2 ̂ 2 / 2 ̂ 2 / 2 ̂ 1 / 2 q 1 Λ ̂ 1 / 2 ̂ 2 / 2 ̂ 2 / 2 ̂ 1 / 2 q 2 Λ ̂ 1 / 2 ) ( B 2 B 1 B 0 B 1 B 2 ) = 0 , q l = ( sin θ + 2 l sin ϕ ) 2 .
E ( x , z ) = m [ ( A m e α m z + A m e α m z ) × l B l , m e 2 π i ( ξ + l b ) x ] .
r + i i = [ n 0 ( 1 + i γ ) ] 2 = n 0 2 n 0 2 γ 2 + 2 i n 0 2 γ .
r = n 0 2 n 0 2 γ 2
i = 2 n 0 2 γ .
E ( z ) = E 0 e i k n 0 ( 1 + i γ ) z , E ( z ) = E 0 e i k n 0 z e k γ n 0 z ,
E ( d ) = E 0 e i k n 0 d e k γ n 0 d ,
T = | E ( d ) E ( 0 ) | 2 = e 2 k γ n 0 d ,
n 0 γ = ln ( T ) 2 k d = 2.3 D 2 k d .
i = n 0 ln ( T ) k d = 2.3 n 0 D k d .
d Δ n = 0.05 λ Δ D .
Q = 2 π λ d n Λ 2 ,
i = ¯ i ( 1 + m cos θ ) ,
T = e k d i / n = e k d ¯ i ( 1 + m cos θ ) / n .
T ¯ = e k d ¯ i / n 2 π π π e k d ¯ i m cos θ / n d θ
T ¯ = e k d ¯ i / n π π 1 2 Γ ( 1 2 ) I 0 ( k d ¯ i m n ) ,
T ¯ = e k d ¯ i / n I 0 ( k d ¯ i m n ) .
T ¯ = exp ( k d ¯ i / n ) ,